To obtain a judgment for damages, you need personal service on the defendants. So you need to locate them in NY and have them served with the summons. Posting the summons and mailing a copy is only effective for obtaining possession, not for money damages.
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Not sure you asked a direct question. I will just give my quick first impression (I am not a FL attty):
You can sue
You can obtain a judgment in FL.
You can move to enforrce your judgment in NY.
t will take some time, some money, and some effort.
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Yes, you can hire a PI to locate the former tenant.
Serve them with the summons from the lawsuit.
Have a hearing in FL on the lawsuit.
Obtain a Final Judgment.
Record a certified copy of the Final Judgment in the jurisdiction where they live.
Commence garnishment or collection efforts in their jurisdiction.
By the way, no tenant can live out their security deposit. It is not legal for landlords or tenants (even if they both "agree") to use the security deposit against charges while the tenant is in the premises. After move-out, the LL may be able to use the security deposit against the outstanding charges if they follow the law. I strongly suggest to my landlord clients that they join a local landlord group to keep up to date on the law. Good luck.